Graph Neural Networks

Graph Neural Networks (GNNs) have emerged as powerful tools for graph learning, and one key challenge arising in GNNs is the development of effective pooling operations for learning meaningful graph representations. In this paper, we propose a novel Edge-Node Attention-based Hierarchical Pooling (ENAHPool) operation for GNNs. Unlike existing cluster-based pooling methods that suffer from ambiguous node assignments and uniform edge-node information aggregation, ENAHPool assigns each node exclusively to a cluster and employs attention mechanisms to perform weighted aggregation of both node features within clusters and edge connectivity strengths between clusters, resulting in more informative hierarchical representations. To further enhance the model performance, we introduce a Multi-Distance Message Passing Neural Network (MD-MPNN) that utilizes edge connectivity strength information to enable direct and selective message propagation across multiple distances, effectively mitigating the over-squashing problem in classical MPNNs. Experimental results demonstrate the effectiveness of the proposed method.

Defending Graph Neural Networks (GNNs) against adversarial attacks requires balancing accuracy and robustness, a trade-off often mishandled by traditional methods like adversarial training that intertwine these conflicting objectives within a single classifier. To overcome this limitation, we propose a self-supervised adversarial purification framework. We separate robustness from the classifier by introducing a dedicated purifier, which cleanses the input data before classification. In contrast to prior adversarial purification methods, we propose GPR-GAE, a novel graph auto-encoder (GAE), as a specialized purifier trained with a self-supervised strategy, adapting to diverse graph structures in a data-driven manner. Utilizing multiple Generalized PageRank (GPR) filters, GPR-GAE captures diverse structural representations for robust and effective purification. Our multi-step purification process further facilitates GPR-GAE to achieve precise graph recovery and robust defense against structural perturbations. Experiments across diverse datasets and attack scenarios demonstrate the state-of-the-art robustness of GPR-GAE, showcasing it as an independent plug-and-play purifier for GNN classifiers. Our code can be found in https://github.com/woodavid31/GPR-GAE.

The expressive power of message-passing graph neural networks (MPNNs) is reasonably well understood, primarily through combinatorial techniques from graph isomorphism testing. However, MPNNs' generalization abilities---making meaningful predictions beyond the training set---remain less explored. Current generalization analyses often overlook graph structure, limit the focus to specific aggregation functions, and assume the impractical, hard-to-optimize $0$-$1$ loss function. Here, we extend recent advances in graph similarity theory to assess the influence of graph structure, aggregation, and loss functions on MPNNs' generalization abilities. Our empirical study supports our theoretical insights, improving our understanding of MPNNs' generalization properties.

Recently, there has been a surge of interest in extending the success of large language models (LLMs) from texts to molecules. Most existing approaches adopt a graph neural network to represent a molecule as a series of node tokens for molecule-language alignment, which, however, have overlooked the inherent hierarchical structures in molecules. Notably, higher-order molecular structures contain rich semantics of functional groups, which encode crucial biochemical functionalities of the molecules. We show that neglecting the hierarchical information in tokenization will lead to subpar molecule-language alignment and severe hallucination. To address this limitation, we propose HIerarchical GrapH Tokenization (HIGHT). HIGHT employs a hierarchical graph tokenizer that encodes the hierarchy of atom, motif, and molecular levels of informative tokens to improve the molecular perception of LLMs. HIGHT also adopts an augmented instruction tuning dataset, enriched with the hierarchical graph information, to further enhance the molecule-language alignment. Extensive experiments on 14 real-world benchmarks verify the effectiveness of HIGHT in reducing hallucination by 40%, and significant improvements in various molecule-language downstream tasks. The project is available at https: //higraphllm.github.io/.

Federated graph learning (FGL) aims to collaboratively train a global graph neural network (GNN) on multiple private graphs with preserving the local data privacy. Besides the common cases of data heterogeneity in conventional federated learning, FGL faces the unique challenge of topology heterogeneity. Most of existing FGL methods alleviate the negative impact of heterogeneity by introducing global signals. However, the manners of creating increments might not be effective and significantly increase the computation amount. In light of this, we propose the FedATH, an FGL method with Alleviating Topology Heterogeneity from a causal perspective. Inspired by the causal theory, we argue that not all edges in a topology are necessary for the training objective, less topology information might make more sense. With the aid of edge evaluator, the local graphs are divided into causal and biased subgraphs. A dual-GNN architecture is used to encode the two subgraphs into corresponding representations. Thus, the causal representations are drawn closer to the training objective while the biased representations are pulled away from it. Further, the Hilbert-Schmidt Independence Criterion is employed to strengthen the separability of the two subgraphs. Extensive experiments on six real-world graph datasets are conducted to demonstrate the superiority of the proposed FedATH over the compared approaches.

Inferring properties of graph-structured data, *e.g.*, the solubility of molecules, essentially involves learning the implicit mapping from graphs to their properties. This learning process is often costly for graph property learners like Graph Convolutional Networks (GCNs). To address this, we propose a paradigm called Graph Nonparametric Teaching (GraNT) that reinterprets the learning process through a novel nonparametric teaching perspective. Specifically, the latter offers a theoretical framework for teaching implicitly defined (*i.e.*, nonparametric) mappings via example selection. Such an implicit mapping is realized by a dense set of graph-property pairs, with the GraNT teacher selecting a subset of them to promote faster convergence in GCN training. By analytically examining the impact of graph structure on parameter-based gradient descent during training, and recasting the evolution of GCNs—shaped by parameter updates—through functional gradient descent in nonparametric teaching, we show *for the first time* that teaching graph property learners (*i.e.*, GCNs) is consistent with teaching structure-aware nonparametric learners. These new findings readily commit GraNT to enhancing learning efficiency of the graph property learner, showing significant reductions in training time for graph-level regression (-36.62\%), graph-level classification (-38.19\%), node-level regression (-30.97\%) and node-level classification (-47.30\%), all while maintaining its generalization performance.

Graph Transformers (GTs) have demonstrated remarkable performance in graph representation learning over popular graph neural networks (GNNs). However, self-attention, the core module of GTs, preserves only low-frequency signals in graph features, leading to ineffectiveness in capturing other important signals like high-frequency ones. Some recent GT models help alleviate this issue, but their flexibility and expressiveness are still limited since the filters they learn are fixed on predefined graph spectrum or spectral order. To tackle this challenge, we propose a Graph Fourier Kolmogorov-Arnold Transformer (GrokFormer), a novel GT model that learns highly expressive spectral filters with adaptive graph spectrum and spectral order through a Fourier series modeling over learnable activation functions. We demonstrate theoretically and empirically that the proposed GrokFormer filter offers better expressiveness than other spectral methods. Comprehensive experiments on 10 real-world node classification datasets across various domains, scales, and graph properties, as well as 5 graph classification datasets, show that GrokFormer outperforms state-of-the-art GTs and GNNs. Our code is available at https://github.com/GGA23/GrokFormer.

As implied by the plethora of literature on graph rewiring, the choice of computational graph employed by a neural network can make a significant impact on its downstream performance. Certain effects related to the computational graph, such as under-reaching and over-squashing, may even render the model incapable of learning certain functions. Most of these effects have only been thoroughly studied in the domain of undirected graphs; however, recent years have seen a significant rise in interest in feedforward computational graphs: directed graphs without any back edges. In this paper, we study the desirable properties of a feedforward computational graph, discovering two important complementary measures: fidelity and mixing time, and evaluating a few popular choices of graphs through the lens of these measures. Our study is backed by both theoretical analyses of the metrics' asymptotic behaviour for various graphs, as well as correlating these metrics to the performance of trained neural network models using the corresponding graphs.

Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields, yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently, intrinsic interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TopInG: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TopInG employs a rationale filtration learning approach to model an autoregressive generating process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TopInG's effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-art methods on both predictive accuracy and interpretation quality.

Graph transformers extend global self-attention to graph-structured data, achieving notable success in graph learning. Recently, random walk structural encoding (RWSE) has been found to further enhance their predictive power by encoding both structural and positional information into the edge representation. However, RWSE cannot always distinguish between edges that belong to different local graph patterns, which reduces its ability to capture the full structural complexity of graphs. This work introduces Simple Path Structural Encoding (SPSE), a novel method that utilizes simple path counts for edge encoding. We show theoretically and experimentally that SPSE overcomes the limitations of RWSE, providing a richer representation of graph structures, particularly in capturing local cyclic patterns. To make SPSE computationally tractable, we propose an efficient approximate algorithm for simple path counting. SPSE demonstrates significant performance improvements over RWSE on various benchmarks, including molecular and long-range graph datasets, achieving statistically significant gains in discriminative tasks. These results pose SPSE as a powerful edge encoding alternative for enhancing the expressivity of graph transformers.

In the kernel density estimation (KDE) problem, we are given a set $X$ of data points in $\mathbb{R}^d$, a kernel function $k: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$, and a query point $\mathbf{q} \in \mathbb{R}^d$, and the objective is to quickly output an estimate of $\sum_{\mathbf{x} \in X} k(\mathbf{q}, \mathbf{x})$. In this paper, we consider $\textsf{KDE}$ in the dynamic setting, and introduce a data structure that efficiently maintains the _estimates_ for a set of query points as data points are added to $X$ over time. Based on this, we design a dynamic data structure that maintains a sparse approximation of the fully connected similarity graph on $X$, and develop a fast dynamic spectral clustering algorithm. We further evaluate the effectiveness of our algorithms on both synthetic and real-world datasets.

Yield of chemical reactions generally depends on the activation barrier, i.e., the energy difference between the reactant and the transition state. Computing the transition state from the reactant and product graphs requires prior knowledge of the correct node alignment (i.e., atom mapping), which is not available in yield prediction datasets. In this work, we propose YieldNet, a neural yield prediction model, which tackles these challenges. Here, we first approximate the atom mapping between the reactants and products using a differentiable node alignment network. We then use this approximate atom mapping to obtain a noisy realization of the condensed graph of reaction (CGR), which is a supergraph encompassing both the reactants and products. This CGR serves as a surrogate for the transition state graph structure. The CGR embeddings of different steps in a multi-step reaction are then passed into a transformer-guided reaction path encoder. Our experiments show that YieldNet can predict the yield more accurately than the baselines. Furthermore, the model is trained only under the distant supervision of yield values, without requiring fine-grained supervision of atom mapping.

Dynamic graph representation learning using Spiking Neural Networks (SNNs) exploits the temporal spiking behavior of neurons, offering advantages in capturing the temporal evolution and sparsity of dynamic graphs. However, existing SNN-based methods often fail to effectively capture the impact of latency in information propagation on node representations. To address this, we propose Delay-DSGN, a dynamic spiking graph neural network incorporating a learnable delay mechanism. By leveraging synaptic plasticity, the model dynamically adjusts connection weights and propagation speeds, enhancing temporal correlations and enabling historical data to influence future representations. Specifically, we introduce a Gaussian delay kernel into the neighborhood aggregation process at each time step, adaptively delaying historical information to future time steps and mitigating information forgetting. Experiments on three large-scale dynamic graph datasets demonstrate that Delay-DSGN outperforms eight state-of-the-art methods, achieving the best results in node classification tasks. We also theoretically derive the constraint conditions between the Gaussian kernel's standard deviation and size, ensuring stable training and preventing gradient explosion and vanishing issues.

We revisit DropEdge, a data augmentation technique for GNNs which randomly removes edges to expose diverse graph structures during training. While being a promising approach to effectively reduce overfitting on specific connections in the graph, we observe that its potential performance gain in supervised learning tasks is significantly limited. To understand why, we provide a theoretical analysis showing that the limited performance of DropEdge comes from the fundamental limitation that exists in many GNN architectures. Based on this analysis, we propose **Aggregation Buffer**, a parameter block specifically designed to improve the robustness of GNNs by addressing the limitation of DropEdge. Our method is compatible with any GNN model, and shows consistent performance improvements on multiple datasets. Moreover, our method effectively addresses well-known problems such as degree bias or structural disparity as a unifying solution. Code and datasets are available at https://github.com/dooho00/agg-buffer.

Quadratic programming (QP) is the most widely applied category of problems in nonlinear programming. Many applications require real-time/fast solutions, though not necessarily with high precision. Existing methods either involve matrix decomposition or use the preconditioned conjugate gradient method. For relatively large instances, these methods cannot achieve the real-time requirement unless there is an effective preconditioner. Recently, graph neural networks (GNNs) opened new possibilities for QP. Some promising empirical studies of applying GNNs for QP tasks show that GNNs can capture key characteristics of an optimization instance and provide adaptive guidance accordingly to crucial configurations during the solving process, or directly provide an approximate solution. However, the theoretical understanding of GNNs in this context remains limited. Specifically, it is unclear what GNNs can and cannot achieve for QP tasks in theory. This work addresses this gap in the context of linearly constrained QP tasks. In the continuous setting, we prove that message-passing GNNs can universally represent fundamental properties of quadratic programs, including feasibility, optimal objective values, and optimal solutions. In the more challenging mixed-integer setting, while GNNs are not universal approximators, we identify a subclass of QP problems that GNNs can reliably represent.

Graphs, fundamental in modeling various research subjects such as computing networks, consist of nodes linked by edges. However, they typically function as components within larger structures in real-world scenarios, such as in protein-protein interactions where each protein is a graph in a larger network. This study delves into the Graph-of-Net (GON), a structure that extends the concept of traditional graphs by representing each node as a graph itself. It provides a multi-level perspective on the relationships between objects, encapsulating both the detailed structure of individual nodes and the broader network of dependencies. To learn node representations within the GON, we propose a position-aware neural network for Graph-of-Net which processes both intra-graph and inter-graph connections and incorporates additional data like node labels. Our model employs dual encoders and graph constructors to build and refine a constraint network, where nodes are adaptively arranged based on their positions, as determined by the network's constraint system. Our model demonstrates significant improvements over baselines in empirical evaluations on various datasets.

Graph neural networks (GNNs) have been widely used in various domains such as social networks, molecular biology, or recommendation systems. Concurrently, different explanations methods of GNNs have arisen to complement its blackbox nature. Explanations of the GNNs’ predictions can be categorized into two types—factual and counterfactual. Given a GNN trained on binary classification into “accept” and “reject” classes, a global counterfactual explanation consists in generating a small set of “accept” graphs relevant to all of the input “reject” graphs. The transformation of a “reject” graph into an “accept” graph is called a recourse. A common recourse explanation is a small set of recourse, from which every “reject” graph can be turned into an “accept” graph. Although local counterfactual explanations have been studied extensively, the problem of finding common recourse for global counterfactual explanation remains unexplored, particularly for GNNs. In this paper, we formalize the common recourse explanation problem, and design an effective algorithm, COMRECGC, to solve it. We benchmark our algorithm against strong baselines on four different real-world graphs datasets and demonstrate the superior performance of COMRECGC against the competitors. We also compare the common recourse explanations to the graph counterfactual explanation, showing that common recourse explanations are either comparable or superior, making them worth considering for applications such as drug discovery or computational biology.

Although directed graphs (digraphs) offer strong modeling capabilities for complex topological systems, existing DiGraph Neural Networks (DiGNNs) struggle to fully capture the concealed rich structural information. This data-level limitation results in model-level sub-optimal predictive performance and underscores the necessity of further exploring the potential correlations between the directed edges (topology) and node profiles (features and labels) from a data-centric perspective, thereby empowering model-centric neural networks with stronger encoding capabilities. In this paper, we propose **E**ntropy-driven **D**igraph knowl**E**dge distillatio**N** (EDEN), which can serve as a data-centric digraph learning paradigm or a model-agnostic hot-and-plug data-centric Knowledge Distillation (KD) module. EDEN implements data-centric machine learning by constructing a coarse-grained Hierarchical Knowledge Tree (HKT) using proposed hierarchical encoding theory, and refining HKT through mutual information analysis of node profiles to guide knowledge distillation during training. As a general framework, EDEN naturally extends to undirected graphs and consistently delivers strong performance. Extensive experiments on 14 (di)graph datasets—spanning both homophily and heterophily settings—and across four downstream tasks show that EDEN achieves SOTA results and significantly enhances existing (Di)GNNs.

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear ``meshing'' PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

Effectively modeling time information and incorporating it into applications or models involving chronologically occurring events is crucial. Real-world scenarios often involve diverse and complex time patterns, which pose significant challenges for time encoding methods. While previous methods focus on capturing time patterns, many rely on specific inductive biases, such as using trigonometric functions to model periodicity. This narrow focus on single-pattern modeling makes them less effective in handling the diversity and complexities of real-world time patterns. In this paper, we investigate to improve the existing commonly used time encoding methods and introduce **Learnable Transformation-based Generalized Time Encoding (LeTE)**. We propose using deep function learning techniques to parameterize nonlinear transformations in time encoding, making them learnable and capable of modeling generalized time patterns, including diverse and complex temporal dynamics. By enabling learnable transformations, LeTE encompasses previous methods as specific cases and allows seamless integration into a wide range of tasks. Through extensive experiments across diverse domains, we demonstrate the versatility and effectiveness of LeTE.